However, Demetrio said, he did not believe the incident was a matter of race.
Dao's daughter Crystal Pepper said at the press conference that her family's "lives have been interrupted" and that "it has been a very difficult time for our entire family, especially my dad, and we are truly grateful for your support."
"What happened to my dad should never happen to any human being, regardless of the circumstance," she added.
United told ABC News that it offered passengers on the flight, from Chicago to Louisville, Kentucky, up to $800 to give up their seats for four crew members who needed to board. No one volunteered, so the airline generated a list of four people to be removed from the flight and reaccommodated, in accordance with the airline's contract of carriage. Three of those people complied. Dao, a doctor with a practice in Louisville, did not, and police were called to remove him.
"The doctor needed to work at the hospital the next day," Twitter user Jayse D. Anspach wrote in a series of posts accompanied by videos of the incident. "So he refused to 'volunteer.'"
"My initial words fell short of truly expressing the shame," he said, adding that the feelings of embarrassment were "palpable" for him and his United colleagues.
United Airlines told ABC News that it is compensating all passengers of the flight for the full price of their tickets.
"All airlines — they need to come out and figure out this overbooking situation," Demetrio said.
"I don't think anybody has to tell any airline, any police force to not do what we all saw on the video," he said. "I think in the short term, that'll never happen again."
Demetrio said the city of Chicago is also responsible for the episode.
One of the officers involved in Dao's forcible removal has been placed on paid administrative leave, pending an investigation, according to the Chicago Department of Aviation. The department said in statement Monday that the incident "was not in accordance with our standard operating procedure, and the actions of the aviation security officer are obviously not condoned by the department."
United released a statement after the press conference, saying, "We continue to express our sincerest apology to Dr. Dao. We cannot stress enough that we remain steadfast in our commitment to make this right. This horrible situation has provided a harsh learning experience, from which we will take immediate, concrete action."
The airline said it would "not ask law enforcement officers to remove passengers from our flights unless it is a matter of safety and security." It also said it was reviewing policies related to the incident.
However, at the press conference, Demetrio said his client had not been contacted by Munoz.
ABC News' James Hill contributed to this report from Chicago. ABC News' Whitney Lloyd and Alex Perez also contributed.